Walking into US composer Elliott Carter's apartment in downtown Manhattan, the first thing you notice is an open-plan study strewn with sheets of manuscript paper, in which a huge desk and a grand piano are almost hidden beneath a flurry of inscrutable sketches and pages of pitches and rhythms. It is the engine room of a life in music, the place that has produced a steady stream of works in recent years, and it's a scene that suggests the energy and productivity of a composer in the prime of his career. Which in a sense, he is - it's just that Carter, who is writing more music than at any time of his life, turned 95 in December. A handful of major composers have had Indian summers - Verdi and Haydn wrote masterpieces in their 70s and 80s - but Carter's tenth decade of creativity is unprecedented in the history of music.

It's an achievement that Carter discusses in disarmingly simple terms. "Maybe I'm instinctive," he says, "but I'm just writing more of what I want to write. It took quite a long time to finally find my vocabulary, and now I just come out with my music without having to think about the problems of musical form, musical harmony, even the presentation of themes. I simply write what I want to hear." In fact, the explosion of Carter's creativity is the hard-won prize of eight decades of continuous compositional labour, in which he has become a living legend in American contemporary music.

His musical journey began with the dawn of modernism. "I was born in 1908, which was the year Schoenberg wrote his Five Orchestral Pieces, and a few years later Stravinsky wrote The Rite of Spring." For almost every other living composer, these pieces have a mythical status as totems of modernist prehistory, but for Carter, they were part of a living musical tradition. "I became interested in music through hearing these new works," he says. "I was very much involved in contemporary music as it was then, which was something that audiences walked out of, because they hated The Rite of Spring. And not far from here" - he surveys the panorama of Manhattan's Lower East Side from his dining room window - "Edgard Varèse used to live, down on Sullivan Street, and I knew him quite well. And I knew Charles Ives."

Varèse's ultramodern music, celebrating the cacophony of 1920s New York with music of mechanistic splendour, was an important inspiration, but Ives, the father of US experimentalism, was even more significant for Carter. Ives sponsored Carter's early musical training, helping him get an interview at Harvard in 1926 with a letter describing him as "rather an exceptional boy", with an "instinctive interest in literature, and especially music, that is somewhat unusual". And in the early 1920s, Ives allowed the young Carter to share his box at Carnegie Hall, firing his interest in the orchestral repertoire he heard played by the Boston Symphony Orchestra.

Carter's compositional challenge was obvious to him. "I had to carry on that particular period, which interested me so much, and meant a great deal to me. I was developing ideas that could be found in composers like Schoenberg, Stravinsky, and Ives." And yet it was only in the early 1950s, with the composition of his First String Quartet, written after a year in the Arizona desert, that he created the musical world that would inspire the rest of his career. The enormous complexity and technical demands of that piece established the uncompromising soundworld that would define his later music. "It took years before anybody could play that quartet effectively," he says.

But even after this discovery, composition was a tortuous process. He completed only eight pieces in the whole of the 1950s and 60s, and today looks back on the works of that period, like the Concerto for Orchestra and Piano Concerto, with incredulity. "My relationship with those pieces is just like one's relationship with one's children," Carter says. (He has one son). "On the whole I like them, but sometimes they worry me - in the sense that when I see them, I don't know how I ever had the patience to write those scores. They have an amazing detail of invention, things that were really very time-consuming. But gradually, it became easier for me to write, as I found what would most interest me to compose."

Carter has lived in America nearly all his life, and in the same apartment since the mid-1940s. But his dogged refusal to bend to the whims of the culture around him makes him a strangely isolated figure in Manhattan. His music is performed much more often in Europe than in America - his only opera, composed in 1998, has never been staged in the US - and he finds himself at odds with the compositional trends that have come and gone in New York.

For example, he was never a fan of Cage's experimentalism. With a typically sardonic chuckle he says: "I think his non-music wears thin very fast. Cage said that all noise is music - well, we've known that from the beginning, except we didn't want to hear it much." He explains the brutal economic realities of his own music's reception in the US. "More than half of my royalties come from Europe, because my music gets played on the radio there, but here, it never does. I mean, this is a commercial radio world - and I don't think my music would sell cat food very well." So does he feel a sense of frustration with his reception in his native country? "It's a situation that I and colleagues of mine in similar situations have become accustomed to," he says. "But fundamentally, I feel that my point of view is American. I think my music is American too, since it describes a sense of discovery, of trying to find a new creative place. I wouldn't want to leave New York now."

His assessment of the future of composition in America is ambivalent. "Young people at Juilliard, where I taught for 20 years, ask me if they should write dissonant music or consonant music. There are some trying to write like Mahler; it seems absurd. However, it's equally absurd for them to try to write like Elliott Carter. But then they can't write like that," he says, almost defensively. "They come to study with me, and they find it's too hard, and they leave. Maybe they're right. But I'm stuck with it."

He faces the future alone: Helen, his wife of 64 years, died last year. "She persuaded me to give up music criticism and compose," he says. "I'm rather messy and confused at times, and she always kept things in order for me. She was very helpful and so kind. I must say I miss her a great deal. But - that's it, you can't do much about that." And yet, instead of indulging in world-weariness or melancholy, or creating a self-consciously "late" style, the pieces he has written in the last year continue to express an astonishing vitality, even youthfulness.

In fact, the only real effect of his advancing age on his music has been to make him work with even greater concentration. "I get tired quickly," he says, "and it's a little bit hard for me to work as much as I used to. And also I get a little annoyed with what I'm doing. In the old days, when I used to get frustrated, I would stop and come back, find out what I should be doing, but I don't like to spend that time any more. As you get older, you don't want to waste it too much."

So too with his music: there are no wasted notes in the transparent textures of his new piece, Dialogues, for piano and ensemble, that the London Sinfonietta will premiere with Nicolas Hodges tonight, a performance Carter will attend. And, from the forest of notes in his study, there will soon emerge a new orchestral work for the Philadelphia Orchestra, a 10-minute piece for Pierre Boulez's 80th birthday, and even, perhaps, a Sixth String Quartet. And who knows - in four years time, the start of a second century of composition.